TOURNAMENT MATRICES WITH EXTREMAL SPECTRAL PROPERTIES

For a tournament matrix M of order n, we define its walk space W(M) to be Span{M(j)1:j = 0,..., n - 1} where 1 is the all ones vector. We show that the dimension of W(M) equals the number of eigenvalues of M whose real parts are greater than -1/2. We then focus on tournament matrices whose walk space has particularly simple structure, and characterize them in terms of their spectra. Specifically, we characterize those tournament matrices such that M(j)I is an eigenvector of M for some j greater-than-or-equal-to 0. We also characterize the tournament matrices M such that J(n) - 2M is a skew-Hadamard matrix. Throughout, we illustrate our results with examples.